Top-dimensional quasiflats in CAT(0) cube complexes

Abstract

We show that every $n$–quasiflat in an $n$–dimensional $CAT\left(0\right)$ cube complex is at finite Hausdorff distance from a finite union of $n$–dimensional orthants. Then we introduce a class of cube complexes, called weakly special cube complexes, and show that quasi-isometries between their universal covers preserve top-dimensional flats. This is the foundational result towards the quasi-isometric classification of right-angled Artin groups with finite outer automorphism group.

Some of our arguments also extend to $CAT\left(0\right)$ spaces of finite geometric dimension. In particular, we give a short proof of the fact that a top-dimensional quasiflat in a Euclidean building is Hausdorff close to a finite union of Weyl cones, which was previously established by Kleiner and Leeb (1997), Eskin and Farb (1997) and Wortman (2006) by different methods.